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At the old entry cohomotopy used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.
(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)
briefly recorded some facts (here) on cohomotopy of 4-manifolds, from Kirby-Melvin-Teichner 12
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added also graphics (here) illustrating the $\mathbb{Z}_2$-equivariant version of the previous example.
Am adding the same illustration also to the respective discussion at equivariant Hopf degree theorem
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H. Sati, U. Schreiber:
Equivariant Cohomotopy implies orientifold tadpole cancellation (arXiv:1909.12277)
where those graphics are taken from
added these references on Cohomotopy cocycle spaces:
Vagn Lundsgaard Hansen, The homotopy problem for the components in the space of maps on the $n$-sphere, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 (DOI:10.1093/qmath/25.1.313)
Vagn Lundsgaard Hansen, On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere, Transactions of the American Mathematical Society Vol. 265, No. 1 (May, 1981), pp. 273-281 (jstor:1998494)
Yes! Eventually we need a higher dimensional table. Or a table of tables.
Personally, of course I am eager to go full blown into twisted equivariant differential Cohomotopy of super orbifolds. And Vincent has a bunch of ideas for what to do. But to make sure not to be barking up the wrong tree, we’d first like to finish one or two further consistency checks in the “topological sector” first.
But that’s just me. If you want to go ahead creating more nLab entries on further variants, please do.
Thanks for the pointer. That made me add also the article that it’s based on:
I think the talk is closer to
so have added that. It seems to rely on values of a function bounded away from $0$ being mapped to a sphere.
Thanks! Interesting. I am adding cross-links with persistent homology (in lack of a general mathematical notion of “persistency” of which these two are examples(?))
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Does this composition amount to some kind of algebra structure on the cohomotopy groups of the spheres with an action?
Sure, composition of $Maps(S^n, S^{n'})$ is the unstable precursor of the graded ring structure on the stable homotopy group of spheres.
Because, by the exchange law (functoriality of the smash product) we may read the product on homotopy groups of spheres in the usual form
$S^{n_1 + n_2} \simeq S^{n_1} \wedge S^{n_2} \overset{ f_1 \wedge f_2 }{ \longrightarrow } S^{n'_1} \wedge S^{n'_2} \simeq S^{ n'_1 + n'_2 }$equivalently as
$S^{n_1 + n_2} \simeq S^{n_1} \wedge S^{n_2} \overset{ f_1 \wedge id_{S^{n_2}} }{ \longrightarrow } S^{n'_1} \wedge S^{n_2} \overset{ id_{S^{n'_1}} \wedge f_2 }{ \longrightarrow } \simeq S^{ n'_1 + n'_2 }$Since, after stabilization, $f_i$ is the same as $f_i \wedge id_{S^k}$, this second version exhibits the product on the homotopy groups of spheres as given by composition of representing maps.
So I guess I’m reaching for some kind of sphere operad, as this question asks at MathOverfow.
By construction, that operad of maps $\underset{i}{\prod} S^{n_i} \longrightarrow S^{n}$ (MO:q/142093) acts on Cohomotopy sets $\pi^{n_i}( X_i )$, by postcomposition of representing maps $X_i \to S^{n_i}$.
There is something slightly more interesting which plays a role for measuring brane charges (writeup upcoming…), namely Cohomotopy classes of spaces which are one-point compactifications of the form
$\big( \mathbb{R}^{b-p} \times S^{9-b} \big)^{cpt} \;\simeq\; S^{b-p} \wedge S^{9-b}_+ \;\simeq\; \big( S^{b-p} \times S^{9-b} \big)/( \{\infty\} \times S^{9-b} )$So these are near-horizon geometries of solitonic $b$-branes in 11d with probe $p$-branes bound to them. They are not quite the plain Cartesian product of spheres, but the result of smashing down at infinite-distance from the $p$-brane the sphere around the $b$-brane.
(An illustration is currently here, but this is temporary and will disappear soon.)
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I hadn’t thought about the case $n=0$ before, but cohomotopy in that degree sends a space to subsets of components.
Thinking negatively, if as in arXiv:math/0608420, p. 14, $S^{-1} = \emptyset$, then $(-1)$-homotopy is as here
The negation of a proposition $\phi$, regarded as the collection of its proofs is $\phi \to \emptyset$. A cocycle $p:\phi \to \emptyset$ is a proof that $\phi$ is false.
Yes. So this means that brane charge measured in equivariant cohomotopy $\pi^V$ sees a special kind of brane sitting at $G$-fixed poins whenever $V^G = 0$, namely a kind of brane of which there is either none or one, but which cannot be superimposed to $N \geq 2$-branes. These “special kind of branes” may be identified (p. 4 here) with O-planes – which, indeed, are supposed to be much like branes, but of which there is either one or none at a given spot.
$\,$
By the way, we submitted our proposal last night. Now I am hitting the road to cross continents in a crazy world. Tomorrow at this time I should be back in the country whose passport I carry. Then I’ll need 24 hours to recover. And then I hope to slowly get back to computing entanglement entropy of the Cayley state.
Added into Examples
Cohomotopy in the lowest dimensions
Since the 0-sphere is the disjoint union of two points, $0$-cohomotopy corresponds to the powerset of the connected components of a space.
Further, the $(-1)$-sphere is understood as the empty space. Since the only map to the empty space is the identity map from the empty space, the $(-1)$-cohomotopy of a space is measuring whether that space is empty. In the context of homotopy type theory, this is the same as negation.
For 0-cohomotopy should one speak of decidable subsets?
Re #41, hope the trip back wasn’t too bad.
added pointer (here and in a couple of related entries) to:
Aravind Asok, Jean Fasel, Mrinal Kanti Das, Euler class groups and motivic stable cohomotopy, Journal of the EMS 24 8 (2022) 2775–2822 [arXiv:1601.05723, doi:10.4171/jems/1156]
Samuel Lerbet, Motivic stable cohomotopy and unimodular rows, Advances in Mathematics 436 109415 (2024) [arXiv:2206.11688, doi:10.1016/j.aim.2023.109415]
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